For a noisy spin system, we derive a nonlocal stochastic version of the overdamped Landau-Lipshitz equation designed to respect the underlying Hamiltonian structure and sample the canonical or Gibbs distribution while being driven by spatially correlated (colored) noise that regularizes the dynamics, making this Stochastic partial differential equation mathematically well-posed. We begin from a microscopic discrete-time model motivated by the Metropolis-Hastings algorithm for a finite number of spins with periodic boundary conditions whose values are distributed on the unit sphere. We thus propose a future state of the system by adding to each spin colored noise projected onto the sphere, and then accept this proposed state with probability given by the ratio of the canonical distribution at the proposed and current states. For uncorrelated (white) noise this process is guaranteed to sample the canonical distribution. We demonstrate that for colored noise, the method used to project the noise onto the sphere and conserve the magnitude of the spins impacts the equilibrium distribution of the system, as coloring projected noise is not equivalent to projecting colored noise. In a specific scenario we show this break in symmetry vanishes with vanishing proposal size; the resulting continuous-time system of Stochastic differential equations samples the canonical distribution and preserves the magnitude of the spins while being driven by colored noise. Taking the continuum limit of infinitely many spins we arrive at the aforementioned version of the overdamped Landau-Lipshitz equation. Numerical simulations are included to verify convergence properties and demonstrate the dynamics.

中文翻译：

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空间相关的噪声驱动自旋系统的非局部随机偏微分方程极限，用于采样规范分布

对于嘈杂的自旋系统，我们推导了过度阻尼的Landau-Lipshitz方程的非本地随机版本，该方程旨在遵循潜在的汉密尔顿结构，并在受到空间相关（有色）噪声的驱动下使规范化，从而规范化或吉布斯分布，从而使动力学规律化。随机偏微分方程在数学上是正确的。我们从Metropolis-Hastings算法激发的微观离散时间模型开始，研究有限数量的自旋，具有周期边界条件，其值分布在单位球体上。因此，我们通过将投影到球体上的每个自旋彩色噪声添加到系统中来提出系统的未来状态，然后以提议状态和当前状态的规范分布之比给出的概率接受此提议状态。对于不相关的（白）噪声，此过程可以保证对规范分布进行采样。我们证明，对于有色噪声，将颜色投射到球体上并保留自旋幅度的方法会影响系统的平衡分布，因为着色投射噪声不等于投射有色噪声。在特定情况下，我们显示此对称性中断随着提议大小的消失而消失。由此产生的随机微分方程的连续时间系统对规范分布进行采样，并在有色噪声驱动下保留自旋的幅度。取无数次自旋的连续极限，我们得出了过度阻尼的Landau-Lipshitz方程的上述形式。